AI 中文总结
研究群的图积的多元增长级数，通过变量代换根据基础图的多元独立多项式推导显式公式，得到RAAGs和RACGs的多元增长级数公式，还表明其系数与图的色多项式有关，弦图系数有完全显式公式。
AI 中文摘要
与有限简单图相关的直角阿廷群（RAAGs）和直角考克斯特群（RACGs）是几何群论中的基本对象。Chiswell经典地用定义图的一元独立多项式通过变量的适当代换来表示它们关于标准生成集的一元增长级数。本文研究群的图积的多元增长级数，并通过变量的适当代换，根据基础图的多元独立多项式推导出显式公式。作为特殊情况，得到了RAAGs和RACGs的多元增长级数公式，扩展了经典的一元恒等式。还表明RAAGs和RACGs的多元增长级数的系数可以用图的双标记和标记色多项式明确描述。这种联系揭示了增长级数和图着色不变量之间丰富的相互作用。特别是，对于弦图的所有系数得到了完全显式的公式，弦图包括树和完全图等。
英文摘要
Right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs) associated with finite simple graphs are fundamental objects in geometric group theory. Their one-variable growth series with respect to the standard generating sets was classically expressed by Chiswell in terms of the one-variable independence polynomial of the defining graph [2] with suitable substitutions of the variable. In this paper, we investigate the multivariate growth series of graph products of groups and derive explicit formulas in terms of the multivariate independence polynomial of the underlying graph through suitable substitutions of variables. As special cases, we obtain multivariate growth series formulas for RAAGs and RACGs, thereby extending the classical one-variable identities. We further show that the coefficients of the multivariate growth series of RAAGs and RACGs admit explicit descriptions in terms of the double-marked and marked chromatic polynomials of graphs. This connection reveals a rich interplay between growth series and graph coloring invariants. In particular, we obtain completely explicit formulas for all the coefficients in the case of chordal graphs, which include, for example, trees and complete graphs.
Comments22 pages
